Order-7 cubic honeycomb

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Order-7 cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,7}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {7}
Vertex figure {3,7}
Order-7 triangular tiling.svg
Dual {7,3,4}
Coxeter group [4,3,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.

Images

Poincaré disk model
Hyperbolic honeycomb 4-3-7 poincare cc.png
Cell-centered
Hyperbolic honeycomb 4-3-7 poincare.png
Order-7 cubic honeycomb cell.png
One cell at center
Order-7 cubic honeycomb cell2.png
One cell with ideal surface

Related polytopes and honeycombs

It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:

It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.

{3,3,7} {4,3,7} {5,3,7} {6,3,7} {7,3,7} {8,3,7} {∞,3,7}
Hyperbolic honeycomb 3-3-7 poincare cc.png Hyperbolic honeycomb 4-3-7 poincare cc.png Hyperbolic honeycomb 5-3-7 poincare cc.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-7 poincare.png Hyperbolic honeycomb i-3-7 poincare.png

Order-8 cubic honeycomb

Order-8 cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,8}
{4,(3,8,3)}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {8}
Vertex figure {3,8}, {(3,4,3)}
H2-8-3-primal.svgH2 tiling 334-4.png
Dual {8,3,4}
Coxeter group [4,3,8]
[4,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has eight cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-8 triangular tiling vertex arrangement.

Hyperbolic honeycomb 4-3-8 poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 4-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of cubic cells.

Infinite-order cubic honeycomb

Infinite-order cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,∞}
{4,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
H2 tiling 23i-4.pngH2 tiling 33i-4.png
Dual {∞,3,4}
Coxeter group [4,3,∞]
[4,((3,∞,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the infinite-order cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,∞}. It has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an infinite-order triangular tiling vertex arrangement.

Hyperbolic honeycomb 4-3-i poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 4-3-i poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cubic cells.

See also

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes
  • Infinite-order hexagonal tiling honeycomb

References

External links